Introduction: After the geometry unit on the area and perimeter of shapes, students are to use this knowledge in figuring out real world problems. This activity will focus on quadrilaterals, circles and their circumference and area of circles. Since this is a rather complex problem, it will involve many steps. Students must understand the difference between an area and a perimeter, and when to use the formulas for the area and the circumference of a circle. It is also important that the students can compare different shapes of the same area. Due to the complexity of the task, students would then be put into groups of 3 or 4 to solve this problem.
Standards:
Common Core Math Standard 7.G.4 Draw, construct and describe geometrical figures and describe the relationships between them. 4. Know the formulas for the area and circumference of a circle and use them to solve problems; give an informal derivation of the relationship between the circumference and area of a circle
Learning target
I will be able to use the formulas for the area and circumference of a circle in order to solve a real world problem.
Material needed:
each group gets a copy of the design
each student gets a copy of the design to cut out
scissors
calculators
paper, pencil
Lesson plan:
Teacher shows a __short video__ on stained glass as an introduction.
Teacher presents a simpler problem, so the students have a general idea of how they would solve the problem that will be given to them: We are making a 1’x1’ stained glass window with 2 different colors. The green glass costs $2.00 per square foot, the yellow glass costs $2.50 per square foot, the frame costs $2 per foot and material to join the glass costs $1.50 per foot. How much will the project cost?
Solution:
Green - $2.00 per sq. ft.
Yellow - $2.50 per sq. ft.
Frame - $2.00 per foot
Inner connection material: $1.50 per foot
There are four sections to the 1’x1’ glass window, so each section would be ¼ (.25) sq ft.
Class brainstorms together and discusses possible strategies.
Discussion questions:
How can we figure out the area of the green glass?
How do we figure out the area of yellow glass?
How do we figure out how much material we need for the frame?
How do we figure out how much material is needed to join the colored panels?
What terms should we understand to make this problem easier to solve?
What formulas could we use to solve this problem?
Students form groups and teacher presents project:
The students in Mr Rivera’s art class are designing a stained glass window for the school’s entry way. The window will be 2 feet tall and 5 feet long. This is what they have designed:They have raised $100.00 for the project. Clear glass is $3.00 per square foot and colored glass is $5.00 per square foot. The material to join the glass together is 10 cents per foot and the frame is $4.00 per foot. Do they have enough money to make the window?
source: http://s3.amazonaws.com/illustrativemathematics/illustration_pdfs/000/001/513/original/illustrative_mathematics_1513.pdf?1372633459
Additional assignment for differentiation:
Based on Seattle, WA sales tax of 9.5%, how much would the amount of sales tax be for the total cost you got from the previous question?
What is the total cost after the sales tax is added?
Teacher monitors group work, provides help and feedback as needed
Groups present their solutions to class. Alternative solutions are discussed.
Solution(s):
Area of window = 2 x 5 = 10 square feet. Area of circles (clear glass) = (10) x 3.14 x .52 = 7.85 ft.2 Area of colored glass = 10 - 7.85 = 2.15 ft.2 At this point, a disconnect between the mathematical solution and the real world should become apparent. Glass is sold in rectangular sheets. Ask the class if 10 one foot circles could be cut from a rectangle 7.85 square foot rectangle. To simplify this problem, ask them what size square would be needed to cut 1 circle 1 foot in diameter. The answer to this would be a 1x1 foot square. This has an area of 1 foot. So to cut 10 circles, they would need a rectangle 1 foot wide by 10 feet long. The students should then see that they must buy at least 10 square feet of clear glass. The cost of the glass would then be: 2.15 x 5 + 10 x 3 = 40.75. To determine how much material is needed for the seams, students must find the circumference of the 10 circles plus the length of the 4 upright sections. 10 x 3.14 x 1 + 4 x 2 =39.4 feet. Multiply by 10 cents a foot to come up with $3.94. The cost of the frame is 4(2+2+5+5) = $56.00. The total cost comes to 56+3.94+ 40.75= $100.69.
Alternative solutions:
Based on Seattle, WA sales tax of 9.5%, how much would the amount of sales tax be for the total cost you got from the previous question?
$100.69 x .095 = $9.57
What is the total cost after the sales tax is added?
$100.69 + $9.57 = $110.26
Assessment:
5 points
3 points
1 point
Number sentence
The math problem is written correctly including +,-,x, / and = sign.
The math problem is mostly written correctly or an incorrect problem is written.
The math problem is not written.
Strategy
The strategy used matches the problem correctly. (area of rectangle and circles, circumference of circles, perimeter of rectangles)
A strategy was used but does not match the problem.
No strategy is used. The strategy does not match the math problem or the answer.
Solution
The answer to the math problem is correct and labeled.
Middle school students are in the process of making the transition from concrete thinking to abstract thinking. Because of this, the use of models and visual aids will help them solve this problem. Besides having a model of the complete window, it would also be helpful for them to make representations (either graphically, drawn with a compass on paper or they could be cut out of a piece of paper) of the 1 foot circles and the squares of material needed to model the squares of glass needed to cut them out of.
Scaffolding
This is a complex real world problem where students need to apply math concepts in order to come up with a solution. Breaking down the problem into smaller steps either as a class or as groups would help the students to understand the problem and would lead them to the solution. As part of this process, students need to apply previously learned word problem strategies such as collecting data, organizing data, analyzed pictures quantitatively.
Reflection and Assessment
Reflection can be done in groups or as a whole class. Reviewing what approach worked and what didn't, builds awareness and analytical thinking. Students may also discuss how their team worked together. Discussions like this might help students to understand group dynamics and their own roles in group settings.
Differentiation
The complexity of this problem allows multiple solutions and therefore provides many great opportunities for differentiation. Also, exploring different scenarios with sales tax and sales allows students more practice with percents and makes this problem even more complex for gifted students.
Modeling Activity
Introduction: After the geometry unit on the area and perimeter of shapes, students are to use this knowledge in figuring out real world problems. This activity will focus on quadrilaterals, circles and their circumference and area of circles. Since this is a rather complex problem, it will involve many steps. Students must understand the difference between an area and a perimeter, and when to use the formulas for the area and the circumference of a circle. It is also important that the students can compare different shapes of the same area. Due to the complexity of the task, students would then be put into groups of 3 or 4 to solve this problem.
- Standards:
Common Core Math Standard 7.G.4Draw, construct and describe geometrical figures and describe the relationships between them.
4. Know the formulas for the area and circumference of a circle and use them to solve problems; give an informal derivation of the relationship between the circumference and area of a circle
- Learning target
I will be able to use the formulas for the area and circumference of a circle in order to solve a real world problem.The students in Mr Rivera’s art class are designing a stained glass window for the school’s entry way. The window will be 2 feet tall and 5 feet long. This is what they have designed:They have raised $100.00 for the project. Clear glass is $3.00 per square foot and colored glass is $5.00 per square foot. The material to join the glass together is 10 cents per foot and the frame is $4.00 per foot. Do they have enough money to make the window?
source: http://s3.amazonaws.com/illustrativemathematics/illustration_pdfs/000/001/513/original/illustrative_mathematics_1513.pdf?1372633459
- Solution(s):
Area of window = 2 x 5 = 10 square feet.Area of circles (clear glass) = (10) x 3.14 x .52 = 7.85 ft.2
Area of colored glass = 10 - 7.85 = 2.15 ft.2
At this point, a disconnect between the mathematical solution and the real world should become apparent. Glass is sold in rectangular sheets. Ask the class if 10 one foot circles could be cut from a rectangle 7.85 square foot rectangle. To simplify this problem, ask them what size square would be needed to cut 1 circle 1 foot in diameter. The answer to this would be a 1x1 foot square. This has an area of 1 foot. So to cut 10 circles, they would need a rectangle 1 foot wide by 10 feet long.
The students should then see that they must buy at least 10 square feet of clear glass. The cost of the glass would then be: 2.15 x 5 + 10 x 3 = 40.75.
To determine how much material is needed for the seams, students must find the circumference of the 10 circles plus the length of the 4 upright sections. 10 x 3.14 x 1 + 4 x 2 =39.4 feet. Multiply by 10 cents a foot to come up with $3.94. The cost of the frame is 4(2+2+5+5) = $56.00. The total cost comes to 56+3.94+ 40.75= $100.69.
- Teaching strategies:
- Modeling and visualisation
Middle school students are in the process of making the transition from concrete thinking to abstract thinking. Because of this, the use of models and visual aids will help them solve this problem. Besides having a model of the complete window, it would also be helpful for them to make representations (either graphically, drawn with a compass on paper or they could be cut out of a piece of paper) of the 1 foot circles and the squares of material needed to model the squares of glass needed to cut them out of.- Scaffolding
This is a complex real world problem where students need to apply math concepts in order to come up with a solution. Breaking down the problem into smaller steps either as a class or as groups would help the students to understand the problem and would lead them to the solution. As part of this process, students need to apply previously learned word problem strategies such as collecting data, organizing data, analyzed pictures quantitatively.- Reflection and Assessment
Reflection can be done in groups or as a whole class. Reviewing what approach worked and what didn't, builds awareness and analytical thinking. Students may also discuss how their team worked together. Discussions like this might help students to understand group dynamics and their own roles in group settings.- Differentiation
The complexity of this problem allows multiple solutions and therefore provides many great opportunities for differentiation.Also, exploring different scenarios with sales tax and sales allows students more practice with percents and makes this problem even more complex for gifted students.